$\dfrac{ 5f + g }{ -9 } = \dfrac{ 6f - 3h }{ 7 }$ Solve for $f$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 5f + g }{ -{9} } = \dfrac{ 6f - 3h }{ 7 }$ $-{9} \cdot \dfrac{ 5f + g }{ -{9} } = -{9} \cdot \dfrac{ 6f - 3h }{ 7 }$ $5f + g = -{9} \cdot \dfrac { 6f - 3h }{ 7 }$ Multiply both sides by the right denominator. $5f + g = -9 \cdot \dfrac{ 6f - 3h }{ {7} }$ ${7} \cdot \left( 5f + g \right) = {7} \cdot -9 \cdot \dfrac{ 6f - 3h }{ {7} }$ ${7} \cdot \left( 5f + g \right) = -9 \cdot \left( 6f - 3h \right)$ Distribute both sides ${7} \cdot \left( 5f + g \right) = -{9} \cdot \left( 6f - 3h \right)$ ${35}f + {7}g = -{54}f + {27}h$ Combine $f$ terms on the left. ${35f} + 7g = -{54f} + 27h$ ${89f} + 7g = 27h$ Move the $g$ term to the right. $89f + {7g} = 27h$ $89f = 27h - {7g}$ Isolate $f$ by dividing both sides by its coefficient. ${89}f = 27h - 7g$ $f = \dfrac{ 27h - 7g }{ {89} }$